A curiosity: the ratio √10 / 3
In the construction, a right triangle with legs 1 and 1/3 keeps appearing; it carries Holden's angle arctan(1/3). Its hypotenuse is √10/3 ≈ 1.05409. As a frequency ratio this is very close to a semitone. This page lets you measure and hear how close, without claiming more than the numbers show.
The numbers
- hypotenuse
- √(1 + 1/9) = √10/3 = 1.0540925534
- its square
- 10/9 exactly
- size in cents
- 600 · log₂(10/9) = 91.20
- angle at E
- arctan(1/3) ≈ 18.43°
A cent is 1/100 of an equal-tempered semitone; most listeners notice differences of about 5 cents or more between two steady tones.
Compared with the semitones of music
| interval | ratio | cents | difference from √10/3 | listen |
|---|
Two facts stand out. The square of √10/3 is exactly 10:9, the just “minor whole tone” of Ptolemy's syntonic diatonic, so √10/3 is exactly half of that tone. And it lies within about 1 cent of the Pythagorean semitone, the limma 256:243, whereas the equal-tempered semitone is almost 9 cents away.
Half a schisma: the exact link to the limma
The “about 1 cent” between √10/3 and the limma has an exact name. Square both sides:
(√10/3)² = 10/9 = (256/243)² × 32805/32768
(check: 65536 × 32805 / (59049 × 32768) = 65610/59049 = 10/9). The last factor, 32805/32768 = 1.954 cents, is the schisma of tuning theory. So √10/3 is the limma widened by exactly half a schisma, 0.977 cents: the minor whole tone 10:9 is two limmas and one schisma.
- √10/3 − limma
- 0.97686 cents = ½ schisma, exactly
- 24 × that gap
- 23.44465 cents = 12 schismas
- Pythagorean comma
- 531441/524288 = 23.46001 cents
- difference
- 0.01536 cents (the “atom of Kirnberger”)
- 3:2 less one schisma
- 16384/10935 = 700.00128 cents (Kirnberger's fifth)
- 12 such fifths − 7 octaves
- 0.01536 cents
Twelve pure fifths 3:2 overshoot seven octaves by the Pythagorean comma, which is why a circle of pure fifths never closes. Twenty-four of these half-schismas (twelve schismas) come within 0.015 cents of that comma, so narrowing each pure fifth by one schisma, that is by (√10/3 ÷ limma)², gives a fifth of 700.0013 cents: equal temperament's fifth to a thousandth of a cent, and a circle that misses closing by only 0.015 cents. You can hear it on The chord wheel (Pentametric circle of fifths, fifth: Kirnberger).
exact the identity is plain arithmetic. known the schisma, the near-equality of twelve schismas and the comma, and the fifth 16384/10935 are classical tuning theory (J. P. Kirnberger, 18th century, proposed this fifth as a whole-number stand-in for equal temperament). What is new here is only the route: from the triangle with legs 1 and 1/3 to that fifth. No fraction of whole numbers can close the circle exactly, since 27/12 is irrational; and √10 only brings square roots, never a 12th root of 2.
Can it replace the equal-tempered semitone?
An equal temperament must close at the octave: twelve equal steps have to make exactly 2:1. Twelve steps of 21/12 give 1200 cents. Twelve steps of √10/3 give only 1094.4 cents, about a semitone short of the octave; it would take 13.16 of them to reach 2:1. So √10/3 cannot be the step of a scale that repeats at the octave. It could be the step of a non-octave scale, a family that composers have explored (for example Wendy Carlos's alpha and beta scales, or the Bohlen–Pierce scale), and that is a legitimate artistic direction rather than a correction of equal temperament.
Status: the closeness of √10/3 to 21/12 is a numerical observation; its distance from the limma is exactly half a schisma (shown above), which ties it to known tuning theory. Observation by the author in GeoGebra, where the triangle has EF = 1, FB3 = 1/3 and EB3 = 1.0540925533894…